Set Of First-order Sentences Research Articles

Dynamic relational mereotopology: Logics for stable and unstable relations

In this paper we present stable and unstable versions of several well-known relations from mereotopology: part-of, overlap, underlap and contact. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereotopo-logical relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for Boolean algebras and distributive lattices. Then we present some results about the first-order predicate logic of these relations and about its quantifier-free fragment. Completeness theorems for these logics are proved, the full first-order theory is proved to be hereditary undecidable and the satisfiability problem of the quantifier-free fragment is proved to be NP-complete.

Logics for stable and unstable mereological relations

Abstract In this paper we present logics about stable and unstable versions of several well-known relations from mereology: part-of, overlap and underlap. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for distributive lattices. First-order predicate logic and modal logic are presented with semantics based on structures with stable and unstable mereological relations. Completeness theorems for these logics are proved, as well as decidability in the case of the modal logic, hereditary undecidability in the case of the first-order logic, and NP-completeness for the satisfiability problem of the quantifier-free fragment of the first-order logic.

Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids

We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In [arXiv:0808.1860v1 [math.LO]] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general.

Configurations in Coproducts of Priestley Spaces

Let P be a configuration, i.e., a finite poset with top element. Let Forb(P) be the class of bounded distributive lat- tices L whose Priestley space P(L) contains no copy of P.W e show that the following are equivalent: Forb(P) is first-order de- finable, i.e., there is a set of first-order sentences in the language of bounded lattice theory whose satisfaction characterizes member- ship in Forb(P); P is coproductive, i.e., P embeds in a coproduct of Priestley spaces iff it embeds in one of the summands; P is a tree. In the restricted context of Heyting algebras, these conditions are also equivalent to ForbH(P) being a variety, or even a quasivariety.

On the Modularization Theorem for logical specifications

Abstract The Modularization Property is a basic tool for guaranteeing the preservation of modular structure under refinements and its importance has been noted by several researchers. In the context of logical specifications, i.e. those presented by sets of first-order sentences of a (possibly many-sorted) language, the role played by the Modularization Property is examined and a proof of the Modularization Theorem is presented.

What should a database know?

The by-now-standard perspective on databases within the deductive database community is that they can be specified by sets of first-order sentences. As such, they can be said to be claims about the truths of some external world; the database is a representation of that world. Within the logical database paradigm, virtually all approaches to database query evaluation treat queries as first-order formulas, usually with free variables whose bindings resulting from the evaluation phase define the answers to the query. Following Levesque [10, 11], we argue that, for greater expressiveness, queries should be formulas in an epistemic modal logic. Queries, in other words, should be permitted to address aspects of the external world as represented by the database, as well as aspects of the database itself, i.e., aspects of what the database knows about that external world. We shall also argue that integrity constraints are best viewed as sentences about what the database knows, not, as is usually the case, as first-order sentences about the external world. On this view, integrity constraints are modal sentences and hence are formally identical to a strict subset of the permissible database queries. Integrity maintenance then becomes formally identical to query evaluation for a certain class of database queries. We formalize these notions in Levesque's language KFOPCE and define the concepts of an answer to a query and of a database satisfying its integrity constraints. We also show that Levesque's axiomatization of KFOPCE provides a suitable logic for reasoning about queries and integrity constraints. Next, we show how to do query evaluation and integrity maintenance for a restricted, but sizable, class of queries/constraints. An interesting feature of this class of queries/constraints is that Prolog's negation as failure mechanism serves to reduce query evaluation to first-order theorem proving. This provides independent confirmation that negation as failure is really an epistemic operator in disguise. We then provide sufficient conditions for the completeness of this query evaluator. Finally, we show how to use this evaluator to answer queries under the closed-world assumption.

On the computability of circumscription

Circumscription has been used to formalize the nonmonotonic aspects of common-sense reasoning. The structure of second-order sentences expressing circumscription for which an equivalent first-order sentence can easily be constructed has been studied by Lifschitz (1985). This paper studies the computability aspects of circumscription. We show that even if the circumscription is expressible as a first-order sentence, it is not computable. We also prove the undecidability of determining whether a given set of first-order sentences has a minimal model, a unique minimal model, a finite number of minimal models or an infinite number of minimal models. In addition, we demonstrate the undecidability of determining if the extension of the minimal predicate is finite or infinite, and whether it is expressible as a first-order sentence.

Nonconvergence, undecidability, and intractability in asymptotic problems

Results delimiting the logical and effective content of asymptotic combinatorics are presented. For the class of binary relations with an underlying linear order, and the class of binary functions, there are properties, given by first-order sentences, without asymptotic probabilities; every first-order asymptotic problem (i.e., set of first-order sentences with asymptotic probabilities bounded by a given rational number between zero and one) for these two classes is undecidable. For the class of pairs of unary functions or permutations, there are monadic second-order properties without asymptotic probabilities; every monadic second-order asymptotic problem for this class is undecidable. No first-order asymptotic problem for the class of unary functions is elementary recursive.

Second-order languages and mathematical practice

There are well-known theorems in mathematical logic that indicate rather profound differences between the logic of first-order languages and the logic of second-order languages. In the first-order case, for example, there is Gödel's completeness theorem: every consistent set of sentences (vis-à-vis a standard axiomatization) has a model. As a corollary, first-order logic is compact: if a set of formulas is not satisfiable, then it has a finite subset which also is not satisfiable. The downward Löwenheim-Skolem theorem is that every set of satisfiable first-order sentences has a model whose cardinality is at most countable (or the cardinality of the set of sentences, whichever is greater), and the upward Löwenheim-Skolem theorem is that if a set of first-order sentences has, for each natural number n, a model whose cardinality is at least n, then it has, for each infinite cardinal κ (greater than or equal to the cardinality of the set of sentences), a model of cardinality κ. It follows, of course, that no set of first-order sentences that has an infinite model can be categorical. Second-order logic, on the other hand, is inherently incomplete in the sense that no recursive, sound axiomatization of it is complete. It is not compact, and there are many well-known categorical sets of second-order sentences (with infinite models). Thus, there are no straightforward analogues to the Löwenheim-Skolem theorems for second-order languages and logic.There has been some controversy in recent years as to whether “second-order logic” should be considered a part of logic, but this boundary issue does not concern me directly, at least not here.